Properties

Label 1110.2.a.k.1.1
Level $1110$
Weight $2$
Character 1110.1
Self dual yes
Analytic conductor $8.863$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1110 = 2 \cdot 3 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1110.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.86339462436\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1110.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +4.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} -1.00000 q^{15} +1.00000 q^{16} +2.00000 q^{17} +1.00000 q^{18} +4.00000 q^{19} +1.00000 q^{20} +4.00000 q^{22} -8.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} -2.00000 q^{26} -1.00000 q^{27} -2.00000 q^{29} -1.00000 q^{30} +8.00000 q^{31} +1.00000 q^{32} -4.00000 q^{33} +2.00000 q^{34} +1.00000 q^{36} +1.00000 q^{37} +4.00000 q^{38} +2.00000 q^{39} +1.00000 q^{40} +10.0000 q^{41} +12.0000 q^{43} +4.00000 q^{44} +1.00000 q^{45} -8.00000 q^{46} -1.00000 q^{48} -7.00000 q^{49} +1.00000 q^{50} -2.00000 q^{51} -2.00000 q^{52} +6.00000 q^{53} -1.00000 q^{54} +4.00000 q^{55} -4.00000 q^{57} -2.00000 q^{58} +4.00000 q^{59} -1.00000 q^{60} -10.0000 q^{61} +8.00000 q^{62} +1.00000 q^{64} -2.00000 q^{65} -4.00000 q^{66} -4.00000 q^{67} +2.00000 q^{68} +8.00000 q^{69} +8.00000 q^{71} +1.00000 q^{72} -6.00000 q^{73} +1.00000 q^{74} -1.00000 q^{75} +4.00000 q^{76} +2.00000 q^{78} -8.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +10.0000 q^{82} -4.00000 q^{83} +2.00000 q^{85} +12.0000 q^{86} +2.00000 q^{87} +4.00000 q^{88} +10.0000 q^{89} +1.00000 q^{90} -8.00000 q^{92} -8.00000 q^{93} +4.00000 q^{95} -1.00000 q^{96} +10.0000 q^{97} -7.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) −1.00000 −0.408248
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) 4.00000 1.20605 0.603023 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) −1.00000 −0.288675
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 1.00000 0.235702
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) 4.00000 0.852803
\(23\) −8.00000 −1.66812 −0.834058 0.551677i \(-0.813988\pi\)
−0.834058 + 0.551677i \(0.813988\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) −2.00000 −0.392232
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) −1.00000 −0.182574
\(31\) 8.00000 1.43684 0.718421 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 1.00000 0.176777
\(33\) −4.00000 −0.696311
\(34\) 2.00000 0.342997
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 1.00000 0.164399
\(38\) 4.00000 0.648886
\(39\) 2.00000 0.320256
\(40\) 1.00000 0.158114
\(41\) 10.0000 1.56174 0.780869 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) 0 0
\(43\) 12.0000 1.82998 0.914991 0.403473i \(-0.132197\pi\)
0.914991 + 0.403473i \(0.132197\pi\)
\(44\) 4.00000 0.603023
\(45\) 1.00000 0.149071
\(46\) −8.00000 −1.17954
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −1.00000 −0.144338
\(49\) −7.00000 −1.00000
\(50\) 1.00000 0.141421
\(51\) −2.00000 −0.280056
\(52\) −2.00000 −0.277350
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) −1.00000 −0.136083
\(55\) 4.00000 0.539360
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) −2.00000 −0.262613
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) −1.00000 −0.129099
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 8.00000 1.01600
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −2.00000 −0.248069
\(66\) −4.00000 −0.492366
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 2.00000 0.242536
\(69\) 8.00000 0.963087
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 1.00000 0.117851
\(73\) −6.00000 −0.702247 −0.351123 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) 1.00000 0.116248
\(75\) −1.00000 −0.115470
\(76\) 4.00000 0.458831
\(77\) 0 0
\(78\) 2.00000 0.226455
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) 10.0000 1.10432
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) 2.00000 0.216930
\(86\) 12.0000 1.29399
\(87\) 2.00000 0.214423
\(88\) 4.00000 0.426401
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 1.00000 0.105409
\(91\) 0 0
\(92\) −8.00000 −0.834058
\(93\) −8.00000 −0.829561
\(94\) 0 0
\(95\) 4.00000 0.410391
\(96\) −1.00000 −0.102062
\(97\) 10.0000 1.01535 0.507673 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) −7.00000 −0.707107
\(99\) 4.00000 0.402015
\(100\) 1.00000 0.100000
\(101\) −18.0000 −1.79107 −0.895533 0.444994i \(-0.853206\pi\)
−0.895533 + 0.444994i \(0.853206\pi\)
\(102\) −2.00000 −0.198030
\(103\) −16.0000 −1.57653 −0.788263 0.615338i \(-0.789020\pi\)
−0.788263 + 0.615338i \(0.789020\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) 20.0000 1.93347 0.966736 0.255774i \(-0.0823304\pi\)
0.966736 + 0.255774i \(0.0823304\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 4.00000 0.381385
\(111\) −1.00000 −0.0949158
\(112\) 0 0
\(113\) −14.0000 −1.31701 −0.658505 0.752577i \(-0.728811\pi\)
−0.658505 + 0.752577i \(0.728811\pi\)
\(114\) −4.00000 −0.374634
\(115\) −8.00000 −0.746004
\(116\) −2.00000 −0.185695
\(117\) −2.00000 −0.184900
\(118\) 4.00000 0.368230
\(119\) 0 0
\(120\) −1.00000 −0.0912871
\(121\) 5.00000 0.454545
\(122\) −10.0000 −0.905357
\(123\) −10.0000 −0.901670
\(124\) 8.00000 0.718421
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 1.00000 0.0883883
\(129\) −12.0000 −1.05654
\(130\) −2.00000 −0.175412
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) −4.00000 −0.348155
\(133\) 0 0
\(134\) −4.00000 −0.345547
\(135\) −1.00000 −0.0860663
\(136\) 2.00000 0.171499
\(137\) 2.00000 0.170872 0.0854358 0.996344i \(-0.472772\pi\)
0.0854358 + 0.996344i \(0.472772\pi\)
\(138\) 8.00000 0.681005
\(139\) 12.0000 1.01783 0.508913 0.860818i \(-0.330047\pi\)
0.508913 + 0.860818i \(0.330047\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 8.00000 0.671345
\(143\) −8.00000 −0.668994
\(144\) 1.00000 0.0833333
\(145\) −2.00000 −0.166091
\(146\) −6.00000 −0.496564
\(147\) 7.00000 0.577350
\(148\) 1.00000 0.0821995
\(149\) −18.0000 −1.47462 −0.737309 0.675556i \(-0.763904\pi\)
−0.737309 + 0.675556i \(0.763904\pi\)
\(150\) −1.00000 −0.0816497
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 4.00000 0.324443
\(153\) 2.00000 0.161690
\(154\) 0 0
\(155\) 8.00000 0.642575
\(156\) 2.00000 0.160128
\(157\) −2.00000 −0.159617 −0.0798087 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) −8.00000 −0.636446
\(159\) −6.00000 −0.475831
\(160\) 1.00000 0.0790569
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) −12.0000 −0.939913 −0.469956 0.882690i \(-0.655730\pi\)
−0.469956 + 0.882690i \(0.655730\pi\)
\(164\) 10.0000 0.780869
\(165\) −4.00000 −0.311400
\(166\) −4.00000 −0.310460
\(167\) 8.00000 0.619059 0.309529 0.950890i \(-0.399829\pi\)
0.309529 + 0.950890i \(0.399829\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 2.00000 0.153393
\(171\) 4.00000 0.305888
\(172\) 12.0000 0.914991
\(173\) −2.00000 −0.152057 −0.0760286 0.997106i \(-0.524224\pi\)
−0.0760286 + 0.997106i \(0.524224\pi\)
\(174\) 2.00000 0.151620
\(175\) 0 0
\(176\) 4.00000 0.301511
\(177\) −4.00000 −0.300658
\(178\) 10.0000 0.749532
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 1.00000 0.0745356
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 0 0
\(183\) 10.0000 0.739221
\(184\) −8.00000 −0.589768
\(185\) 1.00000 0.0735215
\(186\) −8.00000 −0.586588
\(187\) 8.00000 0.585018
\(188\) 0 0
\(189\) 0 0
\(190\) 4.00000 0.290191
\(191\) 8.00000 0.578860 0.289430 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −22.0000 −1.58359 −0.791797 0.610784i \(-0.790854\pi\)
−0.791797 + 0.610784i \(0.790854\pi\)
\(194\) 10.0000 0.717958
\(195\) 2.00000 0.143223
\(196\) −7.00000 −0.500000
\(197\) −26.0000 −1.85242 −0.926212 0.377004i \(-0.876954\pi\)
−0.926212 + 0.377004i \(0.876954\pi\)
\(198\) 4.00000 0.284268
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 1.00000 0.0707107
\(201\) 4.00000 0.282138
\(202\) −18.0000 −1.26648
\(203\) 0 0
\(204\) −2.00000 −0.140028
\(205\) 10.0000 0.698430
\(206\) −16.0000 −1.11477
\(207\) −8.00000 −0.556038
\(208\) −2.00000 −0.138675
\(209\) 16.0000 1.10674
\(210\) 0 0
\(211\) −28.0000 −1.92760 −0.963800 0.266627i \(-0.914091\pi\)
−0.963800 + 0.266627i \(0.914091\pi\)
\(212\) 6.00000 0.412082
\(213\) −8.00000 −0.548151
\(214\) 20.0000 1.36717
\(215\) 12.0000 0.818393
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) −10.0000 −0.677285
\(219\) 6.00000 0.405442
\(220\) 4.00000 0.269680
\(221\) −4.00000 −0.269069
\(222\) −1.00000 −0.0671156
\(223\) −24.0000 −1.60716 −0.803579 0.595198i \(-0.797074\pi\)
−0.803579 + 0.595198i \(0.797074\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) −14.0000 −0.931266
\(227\) −28.0000 −1.85843 −0.929213 0.369546i \(-0.879513\pi\)
−0.929213 + 0.369546i \(0.879513\pi\)
\(228\) −4.00000 −0.264906
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) −8.00000 −0.527504
\(231\) 0 0
\(232\) −2.00000 −0.131306
\(233\) −14.0000 −0.917170 −0.458585 0.888650i \(-0.651644\pi\)
−0.458585 + 0.888650i \(0.651644\pi\)
\(234\) −2.00000 −0.130744
\(235\) 0 0
\(236\) 4.00000 0.260378
\(237\) 8.00000 0.519656
\(238\) 0 0
\(239\) 8.00000 0.517477 0.258738 0.965947i \(-0.416693\pi\)
0.258738 + 0.965947i \(0.416693\pi\)
\(240\) −1.00000 −0.0645497
\(241\) −14.0000 −0.901819 −0.450910 0.892570i \(-0.648900\pi\)
−0.450910 + 0.892570i \(0.648900\pi\)
\(242\) 5.00000 0.321412
\(243\) −1.00000 −0.0641500
\(244\) −10.0000 −0.640184
\(245\) −7.00000 −0.447214
\(246\) −10.0000 −0.637577
\(247\) −8.00000 −0.509028
\(248\) 8.00000 0.508001
\(249\) 4.00000 0.253490
\(250\) 1.00000 0.0632456
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 0 0
\(253\) −32.0000 −2.01182
\(254\) −8.00000 −0.501965
\(255\) −2.00000 −0.125245
\(256\) 1.00000 0.0625000
\(257\) 2.00000 0.124757 0.0623783 0.998053i \(-0.480131\pi\)
0.0623783 + 0.998053i \(0.480131\pi\)
\(258\) −12.0000 −0.747087
\(259\) 0 0
\(260\) −2.00000 −0.124035
\(261\) −2.00000 −0.123797
\(262\) 12.0000 0.741362
\(263\) 8.00000 0.493301 0.246651 0.969104i \(-0.420670\pi\)
0.246651 + 0.969104i \(0.420670\pi\)
\(264\) −4.00000 −0.246183
\(265\) 6.00000 0.368577
\(266\) 0 0
\(267\) −10.0000 −0.611990
\(268\) −4.00000 −0.244339
\(269\) 22.0000 1.34136 0.670682 0.741745i \(-0.266002\pi\)
0.670682 + 0.741745i \(0.266002\pi\)
\(270\) −1.00000 −0.0608581
\(271\) 32.0000 1.94386 0.971931 0.235267i \(-0.0755965\pi\)
0.971931 + 0.235267i \(0.0755965\pi\)
\(272\) 2.00000 0.121268
\(273\) 0 0
\(274\) 2.00000 0.120824
\(275\) 4.00000 0.241209
\(276\) 8.00000 0.481543
\(277\) 22.0000 1.32185 0.660926 0.750451i \(-0.270164\pi\)
0.660926 + 0.750451i \(0.270164\pi\)
\(278\) 12.0000 0.719712
\(279\) 8.00000 0.478947
\(280\) 0 0
\(281\) 10.0000 0.596550 0.298275 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 0 0
\(283\) 12.0000 0.713326 0.356663 0.934233i \(-0.383914\pi\)
0.356663 + 0.934233i \(0.383914\pi\)
\(284\) 8.00000 0.474713
\(285\) −4.00000 −0.236940
\(286\) −8.00000 −0.473050
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) −13.0000 −0.764706
\(290\) −2.00000 −0.117444
\(291\) −10.0000 −0.586210
\(292\) −6.00000 −0.351123
\(293\) 6.00000 0.350524 0.175262 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 7.00000 0.408248
\(295\) 4.00000 0.232889
\(296\) 1.00000 0.0581238
\(297\) −4.00000 −0.232104
\(298\) −18.0000 −1.04271
\(299\) 16.0000 0.925304
\(300\) −1.00000 −0.0577350
\(301\) 0 0
\(302\) −8.00000 −0.460348
\(303\) 18.0000 1.03407
\(304\) 4.00000 0.229416
\(305\) −10.0000 −0.572598
\(306\) 2.00000 0.114332
\(307\) 12.0000 0.684876 0.342438 0.939540i \(-0.388747\pi\)
0.342438 + 0.939540i \(0.388747\pi\)
\(308\) 0 0
\(309\) 16.0000 0.910208
\(310\) 8.00000 0.454369
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 2.00000 0.113228
\(313\) 2.00000 0.113047 0.0565233 0.998401i \(-0.481998\pi\)
0.0565233 + 0.998401i \(0.481998\pi\)
\(314\) −2.00000 −0.112867
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) −18.0000 −1.01098 −0.505490 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) −6.00000 −0.336463
\(319\) −8.00000 −0.447914
\(320\) 1.00000 0.0559017
\(321\) −20.0000 −1.11629
\(322\) 0 0
\(323\) 8.00000 0.445132
\(324\) 1.00000 0.0555556
\(325\) −2.00000 −0.110940
\(326\) −12.0000 −0.664619
\(327\) 10.0000 0.553001
\(328\) 10.0000 0.552158
\(329\) 0 0
\(330\) −4.00000 −0.220193
\(331\) −36.0000 −1.97874 −0.989369 0.145424i \(-0.953545\pi\)
−0.989369 + 0.145424i \(0.953545\pi\)
\(332\) −4.00000 −0.219529
\(333\) 1.00000 0.0547997
\(334\) 8.00000 0.437741
\(335\) −4.00000 −0.218543
\(336\) 0 0
\(337\) 2.00000 0.108947 0.0544735 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) −9.00000 −0.489535
\(339\) 14.0000 0.760376
\(340\) 2.00000 0.108465
\(341\) 32.0000 1.73290
\(342\) 4.00000 0.216295
\(343\) 0 0
\(344\) 12.0000 0.646997
\(345\) 8.00000 0.430706
\(346\) −2.00000 −0.107521
\(347\) 12.0000 0.644194 0.322097 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) 2.00000 0.107211
\(349\) 14.0000 0.749403 0.374701 0.927146i \(-0.377745\pi\)
0.374701 + 0.927146i \(0.377745\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) 4.00000 0.213201
\(353\) 2.00000 0.106449 0.0532246 0.998583i \(-0.483050\pi\)
0.0532246 + 0.998583i \(0.483050\pi\)
\(354\) −4.00000 −0.212598
\(355\) 8.00000 0.424596
\(356\) 10.0000 0.529999
\(357\) 0 0
\(358\) 12.0000 0.634220
\(359\) −8.00000 −0.422224 −0.211112 0.977462i \(-0.567708\pi\)
−0.211112 + 0.977462i \(0.567708\pi\)
\(360\) 1.00000 0.0527046
\(361\) −3.00000 −0.157895
\(362\) −10.0000 −0.525588
\(363\) −5.00000 −0.262432
\(364\) 0 0
\(365\) −6.00000 −0.314054
\(366\) 10.0000 0.522708
\(367\) 8.00000 0.417597 0.208798 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) −8.00000 −0.417029
\(369\) 10.0000 0.520579
\(370\) 1.00000 0.0519875
\(371\) 0 0
\(372\) −8.00000 −0.414781
\(373\) 22.0000 1.13912 0.569558 0.821951i \(-0.307114\pi\)
0.569558 + 0.821951i \(0.307114\pi\)
\(374\) 8.00000 0.413670
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) 4.00000 0.206010
\(378\) 0 0
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 4.00000 0.205196
\(381\) 8.00000 0.409852
\(382\) 8.00000 0.409316
\(383\) 16.0000 0.817562 0.408781 0.912633i \(-0.365954\pi\)
0.408781 + 0.912633i \(0.365954\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −22.0000 −1.11977
\(387\) 12.0000 0.609994
\(388\) 10.0000 0.507673
\(389\) −10.0000 −0.507020 −0.253510 0.967333i \(-0.581585\pi\)
−0.253510 + 0.967333i \(0.581585\pi\)
\(390\) 2.00000 0.101274
\(391\) −16.0000 −0.809155
\(392\) −7.00000 −0.353553
\(393\) −12.0000 −0.605320
\(394\) −26.0000 −1.30986
\(395\) −8.00000 −0.402524
\(396\) 4.00000 0.201008
\(397\) 14.0000 0.702640 0.351320 0.936255i \(-0.385733\pi\)
0.351320 + 0.936255i \(0.385733\pi\)
\(398\) 16.0000 0.802008
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 34.0000 1.69788 0.848939 0.528490i \(-0.177242\pi\)
0.848939 + 0.528490i \(0.177242\pi\)
\(402\) 4.00000 0.199502
\(403\) −16.0000 −0.797017
\(404\) −18.0000 −0.895533
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) 4.00000 0.198273
\(408\) −2.00000 −0.0990148
\(409\) 10.0000 0.494468 0.247234 0.968956i \(-0.420478\pi\)
0.247234 + 0.968956i \(0.420478\pi\)
\(410\) 10.0000 0.493865
\(411\) −2.00000 −0.0986527
\(412\) −16.0000 −0.788263
\(413\) 0 0
\(414\) −8.00000 −0.393179
\(415\) −4.00000 −0.196352
\(416\) −2.00000 −0.0980581
\(417\) −12.0000 −0.587643
\(418\) 16.0000 0.782586
\(419\) −4.00000 −0.195413 −0.0977064 0.995215i \(-0.531151\pi\)
−0.0977064 + 0.995215i \(0.531151\pi\)
\(420\) 0 0
\(421\) −2.00000 −0.0974740 −0.0487370 0.998812i \(-0.515520\pi\)
−0.0487370 + 0.998812i \(0.515520\pi\)
\(422\) −28.0000 −1.36302
\(423\) 0 0
\(424\) 6.00000 0.291386
\(425\) 2.00000 0.0970143
\(426\) −8.00000 −0.387601
\(427\) 0 0
\(428\) 20.0000 0.966736
\(429\) 8.00000 0.386244
\(430\) 12.0000 0.578691
\(431\) 24.0000 1.15604 0.578020 0.816023i \(-0.303826\pi\)
0.578020 + 0.816023i \(0.303826\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 2.00000 0.0961139 0.0480569 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) 0 0
\(435\) 2.00000 0.0958927
\(436\) −10.0000 −0.478913
\(437\) −32.0000 −1.53077
\(438\) 6.00000 0.286691
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 4.00000 0.190693
\(441\) −7.00000 −0.333333
\(442\) −4.00000 −0.190261
\(443\) −12.0000 −0.570137 −0.285069 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) −1.00000 −0.0474579
\(445\) 10.0000 0.474045
\(446\) −24.0000 −1.13643
\(447\) 18.0000 0.851371
\(448\) 0 0
\(449\) 18.0000 0.849473 0.424736 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) 1.00000 0.0471405
\(451\) 40.0000 1.88353
\(452\) −14.0000 −0.658505
\(453\) 8.00000 0.375873
\(454\) −28.0000 −1.31411
\(455\) 0 0
\(456\) −4.00000 −0.187317
\(457\) 18.0000 0.842004 0.421002 0.907060i \(-0.361678\pi\)
0.421002 + 0.907060i \(0.361678\pi\)
\(458\) −10.0000 −0.467269
\(459\) −2.00000 −0.0933520
\(460\) −8.00000 −0.373002
\(461\) 30.0000 1.39724 0.698620 0.715493i \(-0.253798\pi\)
0.698620 + 0.715493i \(0.253798\pi\)
\(462\) 0 0
\(463\) −24.0000 −1.11537 −0.557687 0.830051i \(-0.688311\pi\)
−0.557687 + 0.830051i \(0.688311\pi\)
\(464\) −2.00000 −0.0928477
\(465\) −8.00000 −0.370991
\(466\) −14.0000 −0.648537
\(467\) −12.0000 −0.555294 −0.277647 0.960683i \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) −2.00000 −0.0924500
\(469\) 0 0
\(470\) 0 0
\(471\) 2.00000 0.0921551
\(472\) 4.00000 0.184115
\(473\) 48.0000 2.20704
\(474\) 8.00000 0.367452
\(475\) 4.00000 0.183533
\(476\) 0 0
\(477\) 6.00000 0.274721
\(478\) 8.00000 0.365911
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) −1.00000 −0.0456435
\(481\) −2.00000 −0.0911922
\(482\) −14.0000 −0.637683
\(483\) 0 0
\(484\) 5.00000 0.227273
\(485\) 10.0000 0.454077
\(486\) −1.00000 −0.0453609
\(487\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(488\) −10.0000 −0.452679
\(489\) 12.0000 0.542659
\(490\) −7.00000 −0.316228
\(491\) 4.00000 0.180517 0.0902587 0.995918i \(-0.471231\pi\)
0.0902587 + 0.995918i \(0.471231\pi\)
\(492\) −10.0000 −0.450835
\(493\) −4.00000 −0.180151
\(494\) −8.00000 −0.359937
\(495\) 4.00000 0.179787
\(496\) 8.00000 0.359211
\(497\) 0 0
\(498\) 4.00000 0.179244
\(499\) 4.00000 0.179065 0.0895323 0.995984i \(-0.471463\pi\)
0.0895323 + 0.995984i \(0.471463\pi\)
\(500\) 1.00000 0.0447214
\(501\) −8.00000 −0.357414
\(502\) −12.0000 −0.535586
\(503\) 24.0000 1.07011 0.535054 0.844818i \(-0.320291\pi\)
0.535054 + 0.844818i \(0.320291\pi\)
\(504\) 0 0
\(505\) −18.0000 −0.800989
\(506\) −32.0000 −1.42257
\(507\) 9.00000 0.399704
\(508\) −8.00000 −0.354943
\(509\) −26.0000 −1.15243 −0.576215 0.817298i \(-0.695471\pi\)
−0.576215 + 0.817298i \(0.695471\pi\)
\(510\) −2.00000 −0.0885615
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) −4.00000 −0.176604
\(514\) 2.00000 0.0882162
\(515\) −16.0000 −0.705044
\(516\) −12.0000 −0.528271
\(517\) 0 0
\(518\) 0 0
\(519\) 2.00000 0.0877903
\(520\) −2.00000 −0.0877058
\(521\) −6.00000 −0.262865 −0.131432 0.991325i \(-0.541958\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) −2.00000 −0.0875376
\(523\) −20.0000 −0.874539 −0.437269 0.899331i \(-0.644054\pi\)
−0.437269 + 0.899331i \(0.644054\pi\)
\(524\) 12.0000 0.524222
\(525\) 0 0
\(526\) 8.00000 0.348817
\(527\) 16.0000 0.696971
\(528\) −4.00000 −0.174078
\(529\) 41.0000 1.78261
\(530\) 6.00000 0.260623
\(531\) 4.00000 0.173585
\(532\) 0 0
\(533\) −20.0000 −0.866296
\(534\) −10.0000 −0.432742
\(535\) 20.0000 0.864675
\(536\) −4.00000 −0.172774
\(537\) −12.0000 −0.517838
\(538\) 22.0000 0.948487
\(539\) −28.0000 −1.20605
\(540\) −1.00000 −0.0430331
\(541\) −42.0000 −1.80572 −0.902861 0.429934i \(-0.858537\pi\)
−0.902861 + 0.429934i \(0.858537\pi\)
\(542\) 32.0000 1.37452
\(543\) 10.0000 0.429141
\(544\) 2.00000 0.0857493
\(545\) −10.0000 −0.428353
\(546\) 0 0
\(547\) 20.0000 0.855138 0.427569 0.903983i \(-0.359370\pi\)
0.427569 + 0.903983i \(0.359370\pi\)
\(548\) 2.00000 0.0854358
\(549\) −10.0000 −0.426790
\(550\) 4.00000 0.170561
\(551\) −8.00000 −0.340811
\(552\) 8.00000 0.340503
\(553\) 0 0
\(554\) 22.0000 0.934690
\(555\) −1.00000 −0.0424476
\(556\) 12.0000 0.508913
\(557\) −18.0000 −0.762684 −0.381342 0.924434i \(-0.624538\pi\)
−0.381342 + 0.924434i \(0.624538\pi\)
\(558\) 8.00000 0.338667
\(559\) −24.0000 −1.01509
\(560\) 0 0
\(561\) −8.00000 −0.337760
\(562\) 10.0000 0.421825
\(563\) −12.0000 −0.505740 −0.252870 0.967500i \(-0.581374\pi\)
−0.252870 + 0.967500i \(0.581374\pi\)
\(564\) 0 0
\(565\) −14.0000 −0.588984
\(566\) 12.0000 0.504398
\(567\) 0 0
\(568\) 8.00000 0.335673
\(569\) −22.0000 −0.922288 −0.461144 0.887325i \(-0.652561\pi\)
−0.461144 + 0.887325i \(0.652561\pi\)
\(570\) −4.00000 −0.167542
\(571\) 12.0000 0.502184 0.251092 0.967963i \(-0.419210\pi\)
0.251092 + 0.967963i \(0.419210\pi\)
\(572\) −8.00000 −0.334497
\(573\) −8.00000 −0.334205
\(574\) 0 0
\(575\) −8.00000 −0.333623
\(576\) 1.00000 0.0416667
\(577\) −38.0000 −1.58196 −0.790980 0.611842i \(-0.790429\pi\)
−0.790980 + 0.611842i \(0.790429\pi\)
\(578\) −13.0000 −0.540729
\(579\) 22.0000 0.914289
\(580\) −2.00000 −0.0830455
\(581\) 0 0
\(582\) −10.0000 −0.414513
\(583\) 24.0000 0.993978
\(584\) −6.00000 −0.248282
\(585\) −2.00000 −0.0826898
\(586\) 6.00000 0.247858
\(587\) 12.0000 0.495293 0.247647 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) 7.00000 0.288675
\(589\) 32.0000 1.31854
\(590\) 4.00000 0.164677
\(591\) 26.0000 1.06950
\(592\) 1.00000 0.0410997
\(593\) 10.0000 0.410651 0.205325 0.978694i \(-0.434175\pi\)
0.205325 + 0.978694i \(0.434175\pi\)
\(594\) −4.00000 −0.164122
\(595\) 0 0
\(596\) −18.0000 −0.737309
\(597\) −16.0000 −0.654836
\(598\) 16.0000 0.654289
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) −1.00000 −0.0408248
\(601\) 26.0000 1.06056 0.530281 0.847822i \(-0.322086\pi\)
0.530281 + 0.847822i \(0.322086\pi\)
\(602\) 0 0
\(603\) −4.00000 −0.162893
\(604\) −8.00000 −0.325515
\(605\) 5.00000 0.203279
\(606\) 18.0000 0.731200
\(607\) −8.00000 −0.324710 −0.162355 0.986732i \(-0.551909\pi\)
−0.162355 + 0.986732i \(0.551909\pi\)
\(608\) 4.00000 0.162221
\(609\) 0 0
\(610\) −10.0000 −0.404888
\(611\) 0 0
\(612\) 2.00000 0.0808452
\(613\) −26.0000 −1.05013 −0.525065 0.851062i \(-0.675959\pi\)
−0.525065 + 0.851062i \(0.675959\pi\)
\(614\) 12.0000 0.484281
\(615\) −10.0000 −0.403239
\(616\) 0 0
\(617\) −30.0000 −1.20775 −0.603877 0.797077i \(-0.706378\pi\)
−0.603877 + 0.797077i \(0.706378\pi\)
\(618\) 16.0000 0.643614
\(619\) −4.00000 −0.160774 −0.0803868 0.996764i \(-0.525616\pi\)
−0.0803868 + 0.996764i \(0.525616\pi\)
\(620\) 8.00000 0.321288
\(621\) 8.00000 0.321029
\(622\) 0 0
\(623\) 0 0
\(624\) 2.00000 0.0800641
\(625\) 1.00000 0.0400000
\(626\) 2.00000 0.0799361
\(627\) −16.0000 −0.638978
\(628\) −2.00000 −0.0798087
\(629\) 2.00000 0.0797452
\(630\) 0 0
\(631\) −16.0000 −0.636950 −0.318475 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) −8.00000 −0.318223
\(633\) 28.0000 1.11290
\(634\) −18.0000 −0.714871
\(635\) −8.00000 −0.317470
\(636\) −6.00000 −0.237915
\(637\) 14.0000 0.554700
\(638\) −8.00000 −0.316723
\(639\) 8.00000 0.316475
\(640\) 1.00000 0.0395285
\(641\) 2.00000 0.0789953 0.0394976 0.999220i \(-0.487424\pi\)
0.0394976 + 0.999220i \(0.487424\pi\)
\(642\) −20.0000 −0.789337
\(643\) −28.0000 −1.10421 −0.552106 0.833774i \(-0.686176\pi\)
−0.552106 + 0.833774i \(0.686176\pi\)
\(644\) 0 0
\(645\) −12.0000 −0.472500
\(646\) 8.00000 0.314756
\(647\) 40.0000 1.57256 0.786281 0.617869i \(-0.212004\pi\)
0.786281 + 0.617869i \(0.212004\pi\)
\(648\) 1.00000 0.0392837
\(649\) 16.0000 0.628055
\(650\) −2.00000 −0.0784465
\(651\) 0 0
\(652\) −12.0000 −0.469956
\(653\) −18.0000 −0.704394 −0.352197 0.935926i \(-0.614565\pi\)
−0.352197 + 0.935926i \(0.614565\pi\)
\(654\) 10.0000 0.391031
\(655\) 12.0000 0.468879
\(656\) 10.0000 0.390434
\(657\) −6.00000 −0.234082
\(658\) 0 0
\(659\) −36.0000 −1.40236 −0.701180 0.712984i \(-0.747343\pi\)
−0.701180 + 0.712984i \(0.747343\pi\)
\(660\) −4.00000 −0.155700
\(661\) −2.00000 −0.0777910 −0.0388955 0.999243i \(-0.512384\pi\)
−0.0388955 + 0.999243i \(0.512384\pi\)
\(662\) −36.0000 −1.39918
\(663\) 4.00000 0.155347
\(664\) −4.00000 −0.155230
\(665\) 0 0
\(666\) 1.00000 0.0387492
\(667\) 16.0000 0.619522
\(668\) 8.00000 0.309529
\(669\) 24.0000 0.927894
\(670\) −4.00000 −0.154533
\(671\) −40.0000 −1.54418
\(672\) 0 0
\(673\) 34.0000 1.31060 0.655302 0.755367i \(-0.272541\pi\)
0.655302 + 0.755367i \(0.272541\pi\)
\(674\) 2.00000 0.0770371
\(675\) −1.00000 −0.0384900
\(676\) −9.00000 −0.346154
\(677\) 6.00000 0.230599 0.115299 0.993331i \(-0.463217\pi\)
0.115299 + 0.993331i \(0.463217\pi\)
\(678\) 14.0000 0.537667
\(679\) 0 0
\(680\) 2.00000 0.0766965
\(681\) 28.0000 1.07296
\(682\) 32.0000 1.22534
\(683\) −20.0000 −0.765279 −0.382639 0.923898i \(-0.624985\pi\)
−0.382639 + 0.923898i \(0.624985\pi\)
\(684\) 4.00000 0.152944
\(685\) 2.00000 0.0764161
\(686\) 0 0
\(687\) 10.0000 0.381524
\(688\) 12.0000 0.457496
\(689\) −12.0000 −0.457164
\(690\) 8.00000 0.304555
\(691\) 52.0000 1.97817 0.989087 0.147335i \(-0.0470696\pi\)
0.989087 + 0.147335i \(0.0470696\pi\)
\(692\) −2.00000 −0.0760286
\(693\) 0 0
\(694\) 12.0000 0.455514
\(695\) 12.0000 0.455186
\(696\) 2.00000 0.0758098
\(697\) 20.0000 0.757554
\(698\) 14.0000 0.529908
\(699\) 14.0000 0.529529
\(700\) 0 0
\(701\) 14.0000 0.528773 0.264386 0.964417i \(-0.414831\pi\)
0.264386 + 0.964417i \(0.414831\pi\)
\(702\) 2.00000 0.0754851
\(703\) 4.00000 0.150863
\(704\) 4.00000 0.150756
\(705\) 0 0
\(706\) 2.00000 0.0752710
\(707\) 0 0
\(708\) −4.00000 −0.150329
\(709\) −18.0000 −0.676004 −0.338002 0.941145i \(-0.609751\pi\)
−0.338002 + 0.941145i \(0.609751\pi\)
\(710\) 8.00000 0.300235
\(711\) −8.00000 −0.300023
\(712\) 10.0000 0.374766
\(713\) −64.0000 −2.39682
\(714\) 0 0
\(715\) −8.00000 −0.299183
\(716\) 12.0000 0.448461
\(717\) −8.00000 −0.298765
\(718\) −8.00000 −0.298557
\(719\) 16.0000 0.596699 0.298350 0.954457i \(-0.403564\pi\)
0.298350 + 0.954457i \(0.403564\pi\)
\(720\) 1.00000 0.0372678
\(721\) 0 0
\(722\) −3.00000 −0.111648
\(723\) 14.0000 0.520666
\(724\) −10.0000 −0.371647
\(725\) −2.00000 −0.0742781
\(726\) −5.00000 −0.185567
\(727\) 48.0000 1.78022 0.890111 0.455744i \(-0.150627\pi\)
0.890111 + 0.455744i \(0.150627\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −6.00000 −0.222070
\(731\) 24.0000 0.887672
\(732\) 10.0000 0.369611
\(733\) −34.0000 −1.25582 −0.627909 0.778287i \(-0.716089\pi\)
−0.627909 + 0.778287i \(0.716089\pi\)
\(734\) 8.00000 0.295285
\(735\) 7.00000 0.258199
\(736\) −8.00000 −0.294884
\(737\) −16.0000 −0.589368
\(738\) 10.0000 0.368105
\(739\) −12.0000 −0.441427 −0.220714 0.975339i \(-0.570839\pi\)
−0.220714 + 0.975339i \(0.570839\pi\)
\(740\) 1.00000 0.0367607
\(741\) 8.00000 0.293887
\(742\) 0 0
\(743\) 8.00000 0.293492 0.146746 0.989174i \(-0.453120\pi\)
0.146746 + 0.989174i \(0.453120\pi\)
\(744\) −8.00000 −0.293294
\(745\) −18.0000 −0.659469
\(746\) 22.0000 0.805477
\(747\) −4.00000 −0.146352
\(748\) 8.00000 0.292509
\(749\) 0 0
\(750\) −1.00000 −0.0365148
\(751\) −16.0000 −0.583848 −0.291924 0.956441i \(-0.594295\pi\)
−0.291924 + 0.956441i \(0.594295\pi\)
\(752\) 0 0
\(753\) 12.0000 0.437304
\(754\) 4.00000 0.145671
\(755\) −8.00000 −0.291150
\(756\) 0 0
\(757\) 22.0000 0.799604 0.399802 0.916602i \(-0.369079\pi\)
0.399802 + 0.916602i \(0.369079\pi\)
\(758\) −20.0000 −0.726433
\(759\) 32.0000 1.16153
\(760\) 4.00000 0.145095
\(761\) 10.0000 0.362500 0.181250 0.983437i \(-0.441986\pi\)
0.181250 + 0.983437i \(0.441986\pi\)
\(762\) 8.00000 0.289809
\(763\) 0 0
\(764\) 8.00000 0.289430
\(765\) 2.00000 0.0723102
\(766\) 16.0000 0.578103
\(767\) −8.00000 −0.288863
\(768\) −1.00000 −0.0360844
\(769\) 2.00000 0.0721218 0.0360609 0.999350i \(-0.488519\pi\)
0.0360609 + 0.999350i \(0.488519\pi\)
\(770\) 0 0
\(771\) −2.00000 −0.0720282
\(772\) −22.0000 −0.791797
\(773\) 54.0000 1.94225 0.971123 0.238581i \(-0.0766824\pi\)
0.971123 + 0.238581i \(0.0766824\pi\)
\(774\) 12.0000 0.431331
\(775\) 8.00000 0.287368
\(776\) 10.0000 0.358979
\(777\) 0 0
\(778\) −10.0000 −0.358517
\(779\) 40.0000 1.43315
\(780\) 2.00000 0.0716115
\(781\) 32.0000 1.14505
\(782\) −16.0000 −0.572159
\(783\) 2.00000 0.0714742
\(784\) −7.00000 −0.250000
\(785\) −2.00000 −0.0713831
\(786\) −12.0000 −0.428026
\(787\) −4.00000 −0.142585 −0.0712923 0.997455i \(-0.522712\pi\)
−0.0712923 + 0.997455i \(0.522712\pi\)
\(788\) −26.0000 −0.926212
\(789\) −8.00000 −0.284808
\(790\) −8.00000 −0.284627
\(791\) 0 0
\(792\) 4.00000 0.142134
\(793\) 20.0000 0.710221
\(794\) 14.0000 0.496841
\(795\) −6.00000 −0.212798
\(796\) 16.0000 0.567105
\(797\) −34.0000 −1.20434 −0.602171 0.798367i \(-0.705697\pi\)
−0.602171 + 0.798367i \(0.705697\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 1.00000 0.0353553
\(801\) 10.0000 0.353333
\(802\) 34.0000 1.20058
\(803\) −24.0000 −0.846942
\(804\) 4.00000 0.141069
\(805\) 0 0
\(806\) −16.0000 −0.563576
\(807\) −22.0000 −0.774437
\(808\) −18.0000 −0.633238
\(809\) 26.0000 0.914111 0.457056 0.889438i \(-0.348904\pi\)
0.457056 + 0.889438i \(0.348904\pi\)
\(810\) 1.00000 0.0351364
\(811\) 44.0000 1.54505 0.772524 0.634985i \(-0.218994\pi\)
0.772524 + 0.634985i \(0.218994\pi\)
\(812\) 0 0
\(813\) −32.0000 −1.12229
\(814\) 4.00000 0.140200
\(815\) −12.0000 −0.420342
\(816\) −2.00000 −0.0700140
\(817\) 48.0000 1.67931
\(818\) 10.0000 0.349642
\(819\) 0 0
\(820\) 10.0000 0.349215
\(821\) 30.0000 1.04701 0.523504 0.852023i \(-0.324625\pi\)
0.523504 + 0.852023i \(0.324625\pi\)
\(822\) −2.00000 −0.0697580
\(823\) 16.0000 0.557725 0.278862 0.960331i \(-0.410043\pi\)
0.278862 + 0.960331i \(0.410043\pi\)
\(824\) −16.0000 −0.557386
\(825\) −4.00000 −0.139262
\(826\) 0 0
\(827\) −36.0000 −1.25184 −0.625921 0.779886i \(-0.715277\pi\)
−0.625921 + 0.779886i \(0.715277\pi\)
\(828\) −8.00000 −0.278019
\(829\) 38.0000 1.31979 0.659897 0.751356i \(-0.270600\pi\)
0.659897 + 0.751356i \(0.270600\pi\)
\(830\) −4.00000 −0.138842
\(831\) −22.0000 −0.763172
\(832\) −2.00000 −0.0693375
\(833\) −14.0000 −0.485071
\(834\) −12.0000 −0.415526
\(835\) 8.00000 0.276851
\(836\) 16.0000 0.553372
\(837\) −8.00000 −0.276520
\(838\) −4.00000 −0.138178
\(839\) −24.0000 −0.828572 −0.414286 0.910147i \(-0.635969\pi\)
−0.414286 + 0.910147i \(0.635969\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) −2.00000 −0.0689246
\(843\) −10.0000 −0.344418
\(844\) −28.0000 −0.963800
\(845\) −9.00000 −0.309609
\(846\) 0 0
\(847\) 0 0
\(848\) 6.00000 0.206041
\(849\) −12.0000 −0.411839
\(850\) 2.00000 0.0685994
\(851\) −8.00000 −0.274236
\(852\) −8.00000 −0.274075
\(853\) 54.0000 1.84892 0.924462 0.381273i \(-0.124514\pi\)
0.924462 + 0.381273i \(0.124514\pi\)
\(854\) 0 0
\(855\) 4.00000 0.136797
\(856\) 20.0000 0.683586
\(857\) −6.00000 −0.204956 −0.102478 0.994735i \(-0.532677\pi\)
−0.102478 + 0.994735i \(0.532677\pi\)
\(858\) 8.00000 0.273115
\(859\) −36.0000 −1.22830 −0.614152 0.789188i \(-0.710502\pi\)
−0.614152 + 0.789188i \(0.710502\pi\)
\(860\) 12.0000 0.409197
\(861\) 0 0
\(862\) 24.0000 0.817443
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −2.00000 −0.0680020
\(866\) 2.00000 0.0679628
\(867\) 13.0000 0.441503
\(868\) 0 0
\(869\) −32.0000 −1.08553
\(870\) 2.00000 0.0678064
\(871\) 8.00000 0.271070
\(872\) −10.0000 −0.338643
\(873\) 10.0000 0.338449
\(874\) −32.0000 −1.08242
\(875\) 0 0
\(876\) 6.00000 0.202721
\(877\) −18.0000 −0.607817 −0.303908 0.952701i \(-0.598292\pi\)
−0.303908 + 0.952701i \(0.598292\pi\)
\(878\) 0 0
\(879\) −6.00000 −0.202375
\(880\) 4.00000 0.134840
\(881\) −30.0000 −1.01073 −0.505363 0.862907i \(-0.668641\pi\)
−0.505363 + 0.862907i \(0.668641\pi\)
\(882\) −7.00000 −0.235702
\(883\) 20.0000 0.673054 0.336527 0.941674i \(-0.390748\pi\)
0.336527 + 0.941674i \(0.390748\pi\)
\(884\) −4.00000 −0.134535
\(885\) −4.00000 −0.134459
\(886\) −12.0000 −0.403148
\(887\) −24.0000 −0.805841 −0.402921 0.915235i \(-0.632005\pi\)
−0.402921 + 0.915235i \(0.632005\pi\)
\(888\) −1.00000 −0.0335578
\(889\) 0 0
\(890\) 10.0000 0.335201
\(891\) 4.00000 0.134005
\(892\) −24.0000 −0.803579
\(893\) 0 0
\(894\) 18.0000 0.602010
\(895\) 12.0000 0.401116
\(896\) 0 0
\(897\) −16.0000 −0.534224
\(898\) 18.0000 0.600668
\(899\) −16.0000 −0.533630
\(900\) 1.00000 0.0333333
\(901\) 12.0000 0.399778
\(902\) 40.0000 1.33185
\(903\) 0 0
\(904\) −14.0000 −0.465633
\(905\) −10.0000 −0.332411
\(906\) 8.00000 0.265782
\(907\) −36.0000 −1.19536 −0.597680 0.801735i \(-0.703911\pi\)
−0.597680 + 0.801735i \(0.703911\pi\)
\(908\) −28.0000 −0.929213
\(909\) −18.0000 −0.597022
\(910\) 0 0
\(911\) −40.0000 −1.32526 −0.662630 0.748947i \(-0.730560\pi\)
−0.662630 + 0.748947i \(0.730560\pi\)
\(912\) −4.00000 −0.132453
\(913\) −16.0000 −0.529523
\(914\) 18.0000 0.595387
\(915\) 10.0000 0.330590
\(916\) −10.0000 −0.330409
\(917\) 0 0
\(918\) −2.00000 −0.0660098
\(919\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) −8.00000 −0.263752
\(921\) −12.0000 −0.395413
\(922\) 30.0000 0.987997
\(923\) −16.0000 −0.526646
\(924\) 0 0
\(925\) 1.00000 0.0328798
\(926\) −24.0000 −0.788689
\(927\) −16.0000 −0.525509
\(928\) −2.00000 −0.0656532
\(929\) −14.0000 −0.459325 −0.229663 0.973270i \(-0.573762\pi\)
−0.229663 + 0.973270i \(0.573762\pi\)
\(930\) −8.00000 −0.262330
\(931\) −28.0000 −0.917663
\(932\) −14.0000 −0.458585
\(933\) 0 0
\(934\) −12.0000 −0.392652
\(935\) 8.00000 0.261628
\(936\) −2.00000 −0.0653720
\(937\) 26.0000 0.849383 0.424691 0.905338i \(-0.360383\pi\)
0.424691 + 0.905338i \(0.360383\pi\)
\(938\) 0 0
\(939\) −2.00000 −0.0652675
\(940\) 0 0
\(941\) −42.0000 −1.36916 −0.684580 0.728937i \(-0.740015\pi\)
−0.684580 + 0.728937i \(0.740015\pi\)
\(942\) 2.00000 0.0651635
\(943\) −80.0000 −2.60516
\(944\) 4.00000 0.130189
\(945\) 0 0
\(946\) 48.0000 1.56061
\(947\) 4.00000 0.129983 0.0649913 0.997886i \(-0.479298\pi\)
0.0649913 + 0.997886i \(0.479298\pi\)
\(948\) 8.00000 0.259828
\(949\) 12.0000 0.389536
\(950\) 4.00000 0.129777
\(951\) 18.0000 0.583690
\(952\) 0 0
\(953\) 50.0000 1.61966 0.809829 0.586665i \(-0.199560\pi\)
0.809829 + 0.586665i \(0.199560\pi\)
\(954\) 6.00000 0.194257
\(955\) 8.00000 0.258874
\(956\) 8.00000 0.258738
\(957\) 8.00000 0.258603
\(958\) 24.0000 0.775405
\(959\) 0 0
\(960\) −1.00000 −0.0322749
\(961\) 33.0000 1.06452
\(962\) −2.00000 −0.0644826
\(963\) 20.0000 0.644491
\(964\) −14.0000 −0.450910
\(965\) −22.0000 −0.708205
\(966\) 0 0
\(967\) 32.0000 1.02905 0.514525 0.857475i \(-0.327968\pi\)
0.514525 + 0.857475i \(0.327968\pi\)
\(968\) 5.00000 0.160706
\(969\) −8.00000 −0.256997
\(970\) 10.0000 0.321081
\(971\) −28.0000 −0.898563 −0.449281 0.893390i \(-0.648320\pi\)
−0.449281 + 0.893390i \(0.648320\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) 0 0
\(975\) 2.00000 0.0640513
\(976\) −10.0000 −0.320092
\(977\) 2.00000 0.0639857 0.0319928 0.999488i \(-0.489815\pi\)
0.0319928 + 0.999488i \(0.489815\pi\)
\(978\) 12.0000 0.383718
\(979\) 40.0000 1.27841
\(980\) −7.00000 −0.223607
\(981\) −10.0000 −0.319275
\(982\) 4.00000 0.127645
\(983\) 24.0000 0.765481 0.382741 0.923856i \(-0.374980\pi\)
0.382741 + 0.923856i \(0.374980\pi\)
\(984\) −10.0000 −0.318788
\(985\) −26.0000 −0.828429
\(986\) −4.00000 −0.127386
\(987\) 0 0
\(988\) −8.00000 −0.254514
\(989\) −96.0000 −3.05262
\(990\) 4.00000 0.127128
\(991\) 40.0000 1.27064 0.635321 0.772248i \(-0.280868\pi\)
0.635321 + 0.772248i \(0.280868\pi\)
\(992\) 8.00000 0.254000
\(993\) 36.0000 1.14243
\(994\) 0 0
\(995\) 16.0000 0.507234
\(996\) 4.00000 0.126745
\(997\) −42.0000 −1.33015 −0.665077 0.746775i \(-0.731601\pi\)
−0.665077 + 0.746775i \(0.731601\pi\)
\(998\) 4.00000 0.126618
\(999\) −1.00000 −0.0316386
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1110.2.a.k.1.1 1
3.2 odd 2 3330.2.a.c.1.1 1
4.3 odd 2 8880.2.a.ba.1.1 1
5.4 even 2 5550.2.a.r.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1110.2.a.k.1.1 1 1.1 even 1 trivial
3330.2.a.c.1.1 1 3.2 odd 2
5550.2.a.r.1.1 1 5.4 even 2
8880.2.a.ba.1.1 1 4.3 odd 2